In the anime, you only ever see one version of the same spirit. I remember at the beginning Lucy mentions some of the silver keys can be "Bought in magic shops", but we never see anyone own the same key/spirit as Lucy.
Celestial Spirits in Fairy Tail are generally broken down into Gold and Silver Keys.
For the Gold Keys, there is only one of each spirit, mapped to only one Gold Key. During Lucy's battle against Angel, it was shown that duplicating a key and summoning a Gold Spirit actually desummons them from their current summoner and resummons them under the new summoner, suggesting that indeed there is only one of each.
For the Silver Keys, there are likely multiples of each spirit. Nikola (e.g. Plue) is the most cited example of a family of spirits. In the anime, Caelum is also shown to be owned by multiple summoners at the same time, though this does not constitute canon material as the appearance was in a filler arc.